= Principal ideals in an Artinian local ring with one-dimensional cotangent space
Let $(R,\mathfrak m)$ be an <Artinian local ring>. If $\dim_{R/\mathfrak m}\mathfrak m/\mathfrak m^2=1$, then every ideal of $R$ is principal. The <Nakayama lemma> gives $\mathfrak m=(x)$, and nilpotence gives a finite chain
$$
R\supset(x)\supset(x^2)\supset\cdots\supset0.
$$
For a nonzero ideal $J$, choose the largest $r$ for which $J\subseteq(x^r)$ and take $y\in J\setminus(x^{r+1})$. Writing $y=x^ru$, one has $u\notin\mathfrak m$, so $u$ is a unit and $J=(x^r)=(y)$.
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